In the second part of this talk, I will summarize the results of Bridgeland-Smith and Haiden-Katzarkov-Kontsevich, which are two different, but related, appearances of quadratic differentials in the study of stability conditions. I will then present some results of my research about the spaces of stability conditions on the HKK case, using a definition of relative stability conditions, and time allowing discuss implications of these results for spaces of quadratic differentials.